Chapter 1 Theory of C hoice In these notes, I will summ arize th e basic ideas in choice theory, which you must be familiar with from 14.121. I will d escribe three w ays of modeling individua l c ho ice, nam ely choice fun ctio n, preference, and utility ma ximization. I will present the con-ditions under whic h one can use eac h model. One can alw ays use choice functions in modeling a decision maker’s cho ice at a given situation. In order to represen t a choice function by a comp lete and transitive preference relation, one must ha ve a non-em pty c h oice function that satisfies the w ea k axiom of r evealed preference. Finally, a complete and transferable preference relation can be represen te d b y a utility function, as long as it continuous. 1.1 Alternativ es Consid er a set X of alternativ es. Alternativ es are m utu ally exclusive in the sense that one cannot c h oose two distinct alternativ es at the same tim e. Take also the set of feasible alternatives exhaustiv e so that a decision maker’s c hoices will alwa ys be defined.1 1Note that this is a matter of modeling. For instance, if we have options Coffee and Tea, w e define alternatives as C = Coffee but no Tea, T = Tea but no Coffee, CT = Coffee and Tea, and NT = no Coffee and no Tea. 34 CHAPTER 1. THEORY OF CHOICE 1.2 Choice W hile X consists of all possible altern atives, some of these alternatives m ay no t be feasible for the decision maker. He is constrained to choose from a set A ⊂ X.A choice function describes what a decision maker w o uld have c ho sen under various h y pothetical constraints. Definition 1 A choice function is a mapping c :2X \{∅} → 2X \{∅} such that c (A) ⊆ A for all A ⊆ X. Here, c (A) is meant to be the set of all alternativ es that the decision maker may choose from A. H is actual choice will be a single alternative within c (A).Note that c (A) is non-empt y by definition. In canonical models, it is also assumed that the choice function satisfies the following assumption. Axiom 1 (Weak Axiom of Revealed Preferences) For any A, B X and any ⊆ x, y ∈ A ∩ B,if x ∈ c (A) and y ∈ c (B),then x ∈ c (B). The Weak Axiom of Rev ealed Preferences states that if x is chosen in the presence of y (sothatitisrevealedthat x is at least as good as y), then whenev er y is c ho sen in the presence of x, x could have been chosen, too. This axiom embodies two assump tions. First, the ch oice is a result of binary comparison. Second, the underlying preference used in thecomparison isnot affected by the set A of available alternatives. (For exam ple, the decision maker does not learn from the a vailable cho ices.) 1.3 Preference A relation (on X) is a subset of X × X.A relation º is said to be complete if and only if, given any x, y ∈ X,either x º y or y º x.A relation º is said to be transitive if and only if, giv en any x, y, z ∈ X, [x º y and y º z] ⇒ x º z. Definition 2 Arelation is a preference relation if and only if it is complete and tr an-sitive.1.4. UT ILITY 5 Given an y preference relation º,the strict preference  is defined b y x  y ⇐⇒ [x º y and y º6 x], and the indifference ∼ is defined b y x ∼ y [x º y and y º x].⇐⇒ Here, x º y means that the decision maker finds x at least as good as y; x  y means that the decision maker finds x strictly better than y,and x ∼ y means that the decision maker is indifferen t between x and y. Now, consider a decision maker who c hooses a best alternative according to a pref-erence relation º within each set A ⊆ X of a vailable alternatives. His ch oice function c is given b y º cº (A)={x ∈ A|x º y ∀ y ∈ A} ¡∀ A ∈ 2X \{∅} ¢ . An important question is which choice functions can be thought of coming from suc h a decision maker. This is formulated in the following definition. Definition 3 A choice func tio n c is represented b y º iff c = cº. Repr esentation b y a preference relation º means that decision maker’s choices are mad e as if he tries to choose a best a vailable element according to º. It turns out that the w eak axiom of rev ealed preferences is equivalent to such a representation. Theorem 1 Assum e that X is fin ite. A choice function c is represented by some pref-erence relation º if and only if c satisfies weak axiom of r eveale d pr eferences. It is a useful exercise to show that if c is represen ted by some preference relation º, then it satisfies Axiom 1. For the converse, define ºc by x ºc y ⇐⇒ x ∈ c ({x, y}). Und er Axiom 1, it is anoth er useful exercise to show that c = c.ºc 1.4 Utilit y Arelation º can be represented b y a utility function U : X R in the following sense: → x º y ⇐⇒ U(x) ≥ U(y) ∀ x, y ∈ X. (OR )6 CHAPTER 1. THEORY OF CHOICE The follow ing theorem states that a relation needs to be a preference relation in order to be represented b y a utility functio n . Theorem 2 Assum e that X is finite (or countable). A relation can be presente d by a utility functio n in the sense of (OR ) if and only if it is com p lete and transitive. Moreover, if U : X R represents º,and if f : R R is a strictly increasing function , then → → f U also repr esents º.◦ By the last statement, we call suc h utility representations or dinal.To prove this result for finite X,define U (x)=#{y ∈ X|x º y} and c heck that U represen ts º when º is comple te and transitive. We are mainly intere sted in decision under uncertainty. In that case, the natural set of alternativ es (e.g. the set of all possible lotteries) is infinite. When X is infinite, one also needs to impose a con tinuity assumption. Definition 4 A preference relation º is said to be continuous if and only if the upp er-and lower-contour sets {y|y º x} and {y|x º y} ar e closed for every x ∈ X. Contin uity can also be defined as: for an y two sequences (xn) and (yn) with xn x→ and yn → y, [xn º yn ∀n]=⇒ x …